PSI - Issue 47

Robert Eriksson et al. / Procedia Structural Integrity 47 (2023) 227–237

233

R. Eriksson, A. Azeez / Structural Integrity Procedia 00 (2023) 000–000

7

The plastic zone size, ρ , is determined through a modified strip-yield approach where the material is allowed to harden as ρ increases in size. Mathematically, the procedure involves the steps described below. Loading to P 1 at T 1 results in a WPS plastic zone of size ρ which is obtained by iteratively solving  K 1 = P 1 √ BB N W 2 + 3 2  0 . 886 + 4 . 64 a + ρ W − 13 . 32  a + ρ W  2 + 14 . 72  a + ρ W  3 − 5 . 6  a + ρ W  4 

a + ρ W  1 − a + ρ W 

     

K 1 κ 4 µ √ ρπ

ϵ pz1 = estimated strain σ ys1 = | σ flow ( ϵ pz1 ) | from constitutive model; see Appendix A K ρ 1 = − σ ys1  2( a + ρ ) π    β 1  1 − a a + ρ  1 2 + β 2 3  1 − a a + ρ  3 2 K 1 + K ρ 1 = 0 criterion for stopping iteration

(6)

7 2   

5 

a a + ρ 

7 

a a + ρ 

5 2

β 3

β 4

1 −

1 −

+

+

with respect to ρ (i.e. ρ is increased in increments, starting from ρ = 0, until the condition K 1 + K ρ 1 = 0 is fulfilled). Note that σ flow is taken from the linear isotropic / kinematic hardening model (which must be updated in each iteration as ϵ pz depends on ρ ). K 1 is calculated from Eq. 1 and K ρ 1 fromEq. 3. Changing the load to P 2 results in  K 2 = P 2 √ BB N W 2 + 0 . 886 + 4 . 64 a + ρ W − a + ρ 2 a + ρ 3 a + ρ

3 2 

4 

a + ρ W  1 − a + ρ W 

W 

W 

W 

+ 14 . 72 

− 5 . 6 

13 . 32 

      

K 2 κ 4 µ √ ρπ

estimated strain

ϵ pz2 =

σ 2 =   K ρ 2 = σ 2 

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σ ys2 = | σ flow ( ϵ pz2 ) | if the material plasticize | σ ( ϵ pz2 ) |

see Appendix A

if the material does not plasticize

7 2   

π  

a a + ρ 

3 

a a + ρ 

5 

a a + ρ 

7 

a a + ρ 

 β 1  1 −

1 2

3 2

5 2

2( a + ρ )

β 2

β 3

β 4

1 −

1 −

1 −

+

+

+

which does not need to be solved iteratively as ρ remains the same as in the previous step. Note that the material does not necessarily plasticize during unloading, so the stress σ 2 may take either the value σ ys2 (yield stress) or a stress, σ ( ϵ pz2 ), in the elastic region. However, unloading the crack after the creation of a plastic zone will, at least most of the time, result in some degree of reversed plasticity as the surrounding elastic material forces the plastic zone to shrink. It is here assumed that the crack face pressure introduced by P 1 , σ ys1 , remains at unloading and tries to close the crack. Therefore, after unloading, the plastic zone, ρ , becomes loaded by the positive crack face pressure σ 2 which counteracts the closing pressure σ ys1 . (Reversed plasticity may occur in a zone smaller than the plastic zone, i.e. of size <ρ , however, it should be a fair approximation to assume that σ 2 acts on the full plastic zone ρ .) This loading–unloading procedure will introduce some residual stresses in the plastic zone. The residual stress, σ res is here assumed to simply be the di ff erence between the crack face pressures introduce by P 1 and P 2 , i.e.

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σ res = σ ys1 − σ 2